You have found the following ages (in years) of 6 sloths. Those sloths were randomly selected from the 50 sloths at your local zoo: $ 4,\enspace 19,\enspace 14,\enspace 16,\enspace 6,\enspace 12$ Based on your sample, what is the average age of the sloths? What is the standard deviation? You may round your answers to the nearest tenth.
Solution: Because we only have data for a small sample of the 50 sloths, we are only able to estimate the population mean and standard deviation by finding the sample mean $({\overline{x}})$ and sample standard deviation $({s})$ To find the sample mean , add up the values of all $6$ samples and divide by $6$ $ {\overline{x}} = \dfrac{\sum\limits_{i=1}^{{n}} x_i}{{n}} = \dfrac{\sum\limits_{i=1}^{{6}} x_i}{{6}} $ $ {\overline{x}} = \dfrac{4 + 19 + 14 + 16 + 6 + 12}{{6}} = {11.8\text{ years old}} $ Find the squared deviations from the mean for each sample. Since we don't know the population mean, estimate the mean by using the sample mean we just calculated {60.84} + {51.84} + {4.84} + {17.64} + {33.64} + {0.04}} {{6 - 1}} $ {s^2} = \dfrac{{168.84}}{{5}} = {33.77\text{ years}^2} $ As you might guess from the notation, the sample standard deviation $({s})$ is found by taking the square root of the sample variance $({s^2})$ ${s} = \sqrt{{s^2}}$ $ {s} = \sqrt{{33.77\text{ years}^2}} = {5.8\text{ years}} $ We can estimate that the average sloth at the zoo is 11.8 years old. There is also a standard deviation of 5.8 years.